jmc

You get a digit $0$ on the end of a number whenever it has a factor of $10$, so the question is really asking, how many $10$s are in the prime factorization of $1000!$. Since $10=2\cdot 5$, we need to count how many of each there are. We're going to have more $2$s than $5$s, so we actually only need to count how many times $5$ appears in the prime factorization.

To count how many times a number is divisible by $5$, we divide $1000$ by $5$ to get $200$. Each of those two hundred numbers has a factor of $5$.

Next, how many of the numbers are divisible by $5^2=25$? Dividing $1000$ by $25$, we get $40$. Each of them has two factors of $5$. We've already counted one of them for each number, so for these forty multiples of $25$, we need to add a factor for each to our count.

Next, we need to look at numbers that have $5^3=125$ in them. Eight of our numbers are divisible by $125$, so we count $8$ more factors of $5$.

Finally, we look at $5^4=625$. There is only one number among $1$ through $1000$ divisible by $625$, and that number is $625$, so we only need to count one more factor of $5$. Since $5^5$ is too big, we can stop here.

This gives a total of $200+40+8+1=249$ factors of $5$ in $1000!$, so it has $\boxed{249}$ zeroes on the end.